Dallas Morning News “Anyone who has always loved math for its own sake or for the way it provides new perspectives on important real-world phenomena will find hours of brain-teasing and mind-challenging delight in the British professor's survey of recently answered or still open mathematical questions.... Individual readers will dig deeply into certain chapters and skim others according to personal preference, but every one of them will be captivated by the technical achievements, loose ends and human insights that Stewart shares on his grand mathematical tour.”

New York Journal of Books “Entertaining and accessible.... Ian Stewart belongs to a very small, very exclusive club of popular science and mathematics writers who are worth reading today.” About the Author Ian Stewart is Emeritus Professor of Mathematics and active researcher at the University of Warwick. The author of many books on mathematics, he lives in Coventry, England.

Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers. According to an article by the math historian Kenneth May (Wilson 2014, 2), “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property.”

Three colors are adequate for simpler maps, but an additional fourth color is required for some maps, such as a map in which one region is surrounded by an odd number of other regions that touch each other in a cycle. The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proved in the late 19th century (Heawood 1890); however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852....

“…After more than a decade's efforts using cutting-edge scanning equipment, an international team of scientists has now read about 3,500 characters of explanatory text — a quarter of the original — in the innards of the 2,100-year-old remains.

​They say it was a kind of philosopher's guide to the galaxy, and perhaps the world's oldest mechanical computer. "Now we have texts that you can actually read as ancient Greek, what we had before was like something on the radio with a lot of static," said team member Alexander Jones, a professor of the history of ancient science at New York University. "It's a lot of detail for us because it comes from a period from which we know very little about Greek astronomy and essentially nothing about the technology, except what we gather from here," he said. "So these very small texts are a very big thing for us."

Even the Non-Euclidians Owe Euclid: The King of Infinite Space: Euclid and His Elements by David Berlinski.​Booklist: Starred Review “As dazzling as first love” is how Bertrand Russell described his initial encounter with Euclid. As a mathematician who understands what Russell felt, Berlinski guides his readers through the intellectual wonderland that the ancient Greek geometer created in his epoch-making treatise, the Elements. In writing at once geometrically precise and disarmingly conversational, Berlinski explores the imposing edifice that Euclid erected on a foundation of just five deceptively simple axioms.

Each of these axioms receives careful scrutiny, allowing readers to share in the excitement of mapping out the dimensions of an audacious new human enterprise, inscribing sharp boundaries around key concepts yet opening onto the infinite.

Only an author who thinks both mathematically and philosophically could infer—as Berlinski does—the intellectual and even moral substance of the mental perspective that Euclid unfolds. Readers thus come to realize how Euclid’s pioneering thought made possible the rigor of a mathematical proof—and the discipline of a mathematical life.

Even in the revolutionary modern theorizing of non-Euclidian geometers such as Lobachevsky, Bolyai, and Poincaré, readers will discern Euclid’s abiding influence as a visionary who glimpsed the mathematical unities hidden beneath chaotic appearances.

The tablet contains four columns and 15 rows of cuneiform numbers, which conform to the Pythagorean theorem — the relationship between three sides of a right triangle. Over the years, researchers have theorized that the tablet was evidence of the use of trigonometry, while others have suggested that the tablet might have been mathematical exercises used by a teacher. This new study claims that the tablet could be evidence “of a completely unfamiliar kind and was ahead of its time by thousands of years.”

Mansfield and Wildberger say that if their interpretation is correct, Plimpton 322 would not only be the oldest known trigonometric table, but it would also be the “world’s only completely accurate trigonometric table.” The study’s authors note that this form of trigonometry is different from what’s used today: it wouldn’t use angles or approximations, because that base-60 system would allow mathematicians to use whole numbers, leading to exact calculations, which in turn would be useful for constructing fields, canals, or buildings.

The theory is not without some criticism, according to Science. Historian Mathieu Ossendrijver of Humboldt University in Berlin notes that there’s no proof that the Babylonians used this tablet for construction, while mathematical historian Christine Proust of the French National Center for Scientific Research in Paris says that while the idea makes sense, it’s “highly speculative.”

,700-year-old clay tablet has proven that the Babyloniansdeveloped trigonometry 1,500 years before the Greeks and were using a sophisticated method of mathematics which could change how we calculate today.

The tablet, known as Plimpton 332, was discovered in the early 1900s in Southern Iraq by the American archaeologist and diplomat Edgar Banks, who was the inspiration for Indiana Jones.

The true meaning of the tablet has eluded experts until now but new research by the University of New South Wales, Australia, has shown it is the world’s oldest and most accurate trigonometric table, which was probably used by ancient architects to construct temples, palaces and canals.

However unlike today’s trigonometry, Babylonian mathematics used a base 60, or sexagesimal system, rather than the 10 which is used today. Because 60 is far easier to divide by three, experts studying the tablet, found that the calculations are far more accurate.

“Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles,” said Dr Daniel Mansfield of the School of Mathematics and Statistics in the UNSW Faculty of Science.“It is a fascinating mathematical work that demonstrates undoubted genius. The tablet not only contains the world’s oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.

“This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3000 years, but it has possible practical applications in surveying, computer graphics and education.

“This is a rare example of the ancient world teaching us something new.”

The Greek astronomer Hipparchus, who lived around 120BC, has long been regarded as the father of trigonometry, with his ‘table of chords’ on a circle considered the oldest trigonometric table.

A trigonometric table allows a user to determine two unknown ratios of a right-angled triangle using just one known ratio. But the tablet is far older than Hipparchus, demonstrating that the Babylonians were already well advanced in complex mathematics far earlier.

The tablet, which is thought to have come from the ancient Sumerian city of Larsa, has been dated to between 1822 and 1762 BC. It is now in the Rare Book and Manuscript Library at Columbia University in New York.

“Plimpton 322 predates Hipparchus by more than 1000 years,” says Dr Wildberger.“It opens up new possibilities not just for modern mathematics research, but also for mathematics education. With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own.

“A treasure-trove of Babylonian tablets exists, but only a fraction of them have been studied yet. The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us.”

The 15 rows on the tablet describe a sequence of 15 right-angle triangles, which are steadily decreasing in inclination.

The left-hand edge of the tablet is broken but the researchers believe t there were originally six columns and that the tablet was meant to be completed with 38 rows.

“Plimpton 322 was a powerful tool that could have been used for surveying fields or making architectural calculations to build palaces, temples or step pyramids,” added Dr Mansfield.​The new study is published in Historia Mathematica, the official journal of the International Commission on the History of Mathematics.

Archimedes of Syracuse was an ancient astronomer, war planner, engineer, astrologer and mathematician whose works are mostly lost because of the tragic fire Cleopatra set to the Library of Alexandria, a ruse permitting her time to drown her baby brother, a usurper of Macedonian rule in Polemic Egypt. The fire destroyed the libraries collection of ancient works only known to us today by reference from surviving relics or papyri.

In the early 20th century, researchers in Istanbul discovered Archimedes' work re-written on a prayer book.

This is story of the ancient world's greatest scientific mind: Archimedes of Syracuse.

"The insides of this book are as clever and compelling as the subtitle on the cover. Havil, a retired former master at Winchester College in England, where he taught math for decades, takes readers on a history of irrational numbers--numbers, like v2 or p, whose decimal expansion 'is neither finite nor recurring.' We start in ancient Greece with Pythagoras, whose thinking most likely helped to set the path toward the discovery of irrational numbers, and continue to the present day, pausing to ponder such questions as, 'Is the decimal expansion of an irrational number random?'"--Anna Kuchment, Scientific American

"The Irrationals is a true mathematician's and historian's delight."--Robert Schaefer, New York Journal of Books

"From its lively introduction straight through to a rousing finish this is a book which can be browsed for its collection of interesting facts or studied carefully by anyone with an interest in numbers and their history. . . . This is a wonderful book which should appeal to a broad audience. Its level of difficulty ranges nicely from ideas accessible to high school students to some very deep mathematics. Highly recommended!"--Richard Wilders, MAA Reviews

"To follow the mathematical sections of the book, the reader should have at least a second-year undergraduate mathematical background, as the author does not shrink from providing some detailed arguments. However, the presentation of historical material is given in modern mathematical form. Many readers will encounter unfamiliar and surprising material in this field in which much remains to be explored."--E. J. Barbeau, Mathematical Reviews Clippings

"[I]t is a book that can be warmly recommended to any mathematician or any reader who is generally interested in mathematics. One should be prepared to read some of the proofs. Skipping all the proofs would do injustice to the concept, leaving just a skinny skeleton, but skipping some of the most advanced ones is acceptable. The style, the well documented historical context and quotations mixed with references to modern situations make it a wonderful read."--A. Bultheel, European Mathematical Society​"This is a well-written book to which senior high school students who do not intend to study mathematics at university should be exposed in their last two years at school. The ideas are challenging and provocative, with numerous clear diagrams. The topics are presented with numerous examples, and unobtrusive humour which renders the exposition even more palatable. The book would be an ideal source of ideas in a mathematics course within a liberal arts college because it links not only with the historical source of mathematics problems, but also with some of the great ideas of philosophy."--A. G. Shannon, Notes on Number Theory and Discrete Mathematics

Hitler's ascent to power in 1933 marked a profound change in the balance of power in mathematics. Not only did German-Jewish mathematicians escape to Britain and America, but a revival of French mathematics began under the influence of young scholars who had gone to Germany to study. Lamenting the lack of suitable modern textbooks, they created their own using the pseudonym Nicolas Bourbaki, the name of a French General in the Franco-Prussian War.

The Greek origin of the name suited the intended style of a multi-volume work under the heading 'Elements of Mathematics', inspired by Euclid's previous ancient work. Following their first meeting in 1934, the Bourbaki group's influence went international, with new volumes of 'Elements' published over 50 years. After WWII, they produced a series of advanced monographs by individual authors based on public lectures given from the Sorbonne in Paris, the new emerging center for mathematics.

During this time a brilliant young boy named Alexander Grothendieck (pronounced Groth-then-deck), born in Berlin in 1928, was hidden on a farm in northern Germany. His father, Alexander Schapiro, a Russian anarchist, had taken part in the failed 1905 Russian Revolution. He lost an arm escaping from prison ten years later; leaving Russia on forged papers, he reached Berlin in 1922. When the Nazi's took power in 1933 Schapiro escaped to France and was later joined by Alexander's mother, Hanka Grothendieck. Both parents joined the effort that became the Spanish Civil War. After returning to France in 1939, Schapiro was interned in Camp Vernet in the French Pyrenees, before the Vichy regime handed him over to the Nazi's upon disappearing into Auschwitz. That year, Alexander's mother decided to leave Germany, putting her son Alexander on a train to France, where he would later join his mother, living in various camps for displaced persons.

After the war, Alexander Grothendieck became a student at Montpellier. There his professor told him that a previous protege named Henri Lebesgue, a French mathematician who died in 1941, had already solved all the problems of mathematics but his ideas and methods were too difficult to teach. Undeterred, Alexander Grothendieck rediscovered much of Lebesgue's great work. He achieved his first mathematical success in isolation, honestly believing that he was the only mathematician in the world.

Alexander was a phenomenon, unique by any standard. After writing a dense and profound article that fueled the research of a whole school of scholarly thought, he abandoned the discipline altogether. He late said of himself that he was destined to be the builder of houses he would never inhabit. Aged 27, he started a new chapter in his life, turning the subject of algebraic geometry into something far grander.

Roughly speaking, algebraic geometry studies curves and surfaces representing solutions to algebraic equations. For example, solutions to the equation X+Y=Z determine a surface in three dimensional space framed by the coordinates x,y,z. This particular equation inspired Fermat's last theorem, which states that when (n, meaning any number) is greater than 2 there are no non-trivial solutions for which x,y,z are whole numbers. In other works, the surface defined by that equation contains no such points.

Fermat's famous conjecture, remained unresolved for more then 350 years until Alexander Grothendieck tackled it. Its resolution is extraordinarily difficult: direct number-theoretic approaches never succeeded, nor did direct geometric approaches. Its solution finally came from a viewpoint at a far higher level of abstraction.

His experiences, hidden in Germany for years, then escaping to France, losing a father who battled Tsarist Russia and the Communists and who was finally murdered by the Nazi's, gave Alexander a yearning for extreme abstraction. He would have nothing to do with physics, nor with any kind of military support for mathematics. When he resided at the infamous Institue des Hautes Etudes Scientifiques in Paris he discovered that some of his work was defense related he abandoned that brilliant research center.

In 1970, he went to Montpellier, when he had once been a student. It was the beginning of the end of his mathematical work, and after retirement from academia he went to live in the French Pyrenees, not far from the internment camp where his father had lived before deportation to Auschwitz. Grothendieck's address and telephone number were known only to a select few, sworn to secrecy.

In happier days, the Bourbaki group had been ready and willing to help Alexander. In particular, Jean Dieudonne and another mathematician continued to encourage him to write and tackle contemporary mathematical problems. Alexander began his magnum opus containing all his life's work. With the help of his 12 personal disciples, Alexander Grothendieck's magnum opus on algebraic geometry spanned ten thousand pages.

Like his mathematical predecessors, Gauss and Riemann, and the physicist Einstein, Grothendieck was fascinated by the concept of space. For him, the key ingredient to his work remained the concept of a 'point', something resembling Euclid's notion of something having no dimension. He drew algebraic geometry into a broader context embracing not just curves and surfaces but much of number theory, creating highly sophisticated mathematical tools to handle his specific abstract terrain.

It departed from the usual more concrete concerns about equations and their geometric representations. Eventually Grothendieck himself departed from mundane concerns altogether. He abandoned his best disciples, his five children from three different mothers, and reached for ever deeper abstraction. This man, who eschewed war, gave mathematical seminars to North Vietnamese during U.S. bombing raids! He fought long custody battles for his eldest son and a later court case involving an enormous home he built on his mothers property. ​ He would finally return to his Montpellier home, retreat under the Pyrenees and write theology.

Alexander Grothendieck, died in November 2014 at the age of 86.

His commentaries along with his magnum opus reveal a contemporary Pythagoras, known to those who can understand him.

NOTE: Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety.

When Greek Athenians mastered the social, political challenge of moving from a subsistence based farming economy to an economy based on exports, a political economy that mastered the challenge of colonial maritime trade, Athenians experienced a level of leisure, excellence and achievement whose impact is still felt today.

The Greek maritime colony in Sicily is where the great predecessors of Plato and Aristotle lived, they were thinkers who established schools of thought that demonstrated profound complexity, especially mathematically. It was in the realm of philosophical rationalism that a man named Zeno (of Elea, because people's last names where assigned to them after great achievements which usually took the name of one's home city).

​Zeno's paradox is a conception, a pure idea of infinity that cannot be empirically reconciled. It is thought divorced from the created order, it is numerical denotation to infinity. For Zeno and his followers applying it empirically demonstrates an unsolvable realm, a paradox. Let's examine it. Keep in mind that Zeno and his followers thought of dividing every conceivable number, measurement or digit by half, effectively demonstrating that space and its numerical equivalent are infinite.

Zeno's paradox is explained as involving a fast runner and slow tortoise (turtle), if you place the tortoise ahead at the runner, the runner cannot ever catch up because the runner continually needs to reach half the remaining distance infinitely.

For modern people space, time and measurement are discrete and definite so we don't experience Zeno's paradox. Non-Eucledian mathematics, especially prime numbers (a methodical approach of mathematical thought similar to Zeno's school) is applied by government agencies today as encryption.

Although Aristotle and his followers conceived of applied mathematics empirically, his predecessor Zeno applied mathematics as a 'thought experiment', of mathematical ideas, relations, angles conceived in thought without experiment. Zeno's school of thought provided ample cover for numerous philosophies in Athens, especially cynicism and Epicureanism.

Ancient Incas used Quipus (pronounced key-poos) as both numerical notation, calendars and historical memory banks. With Quipus, Incan Civilization demonstrated a unique unity of thought and feeling that would remain fragmented for westerners until the rise of the internet. Non western civilizations had superior haptic sense, desiring a synoptic view of indigenous life.

​Until 2005, the oldest quipus was dated at 650 A.D., until a westerner visiting the Incan city of Caral found an ancient quipus dated about 3,000 B.C.

Incan civilization didn't have extensive developed writing, nor papyrus or libraries. Using woven cloth and knots they reordered events and denoted logical numerical systems for reference. Knot types, positions, cord directions and levels all corresponded with color and spacing. Quipus needed interpretation so elaborate schools were developed to teach interpretation.

Quipus now dispels western hierarchies that civilizations must develop writing before mathematics! Civilizations can reach complex levels of social, material organization without written records.